The Klein View of Geometry 
In euclidean geometry, for two points A and A', there are elements of E(2) which map A to A'. A translation is one possibility. In our other geometries, a similar result holds. In the similarity and inversive geometries, this is easy to see. In hyperbolic geometry, it is a consequence of the Interchange Lemma.
However, in euclidean geometry, if L = (A,B) and L' = (A',B') are lists of distinct points, then In similarity geometry, we can map pairs to pairs. We shall call this


The Fundamental Theorem of Similarity Geometry If L = (A,B) and L' = (A',B') are lists of distinct points of E, then there is a unique element of S^{+}(2) which maps L to L'.
We can go no further since similarity geometry as the property of angle. Thus we can map


In inversive geometry, the situation is more interesting. We have The Fundamental Theorem of Inversive Geometry
If L = (A,B,C) and L' = (A',B',C') are lists of distinct points of E^{+}, then This result and its proof appear on the inversive group pages.
In hyperbolic geometry, we have the property of hyperbolic distance, so the situation is similar
The other geometries we consider, affine and projective, have the property that two distinct points The Fundamental Theorem of Affine Geometry
If L = (A,B,C) and L' = (A',B',C') are lists of noncollinear points of E, then A proof can be found in the affine group pages.
Since affine transformations preserve ratio along a line, we cannot in general map The Fundamental Theorem of Projective Geometry
If L = (A,B,C,D) and L' = (A',B',C',D) are lists of points of RP^{2}, each with no three collinear, then A proof can be found in the projective group pages.
Since projective transformations preserve the crossratio of four collinear ppoints, we cannot 
